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Chapter 16: Problem 64

What is the value of a googolplex?

Short Answer

Expert verified
A googolplex is \(10^{10^{100}}\), an unimaginably large number.

Step by step solution

01

Define a Googol

A googol is a mathematical term that represents a very large number. Specifically, it is equivalent to \(10^{100}\), which is the digit 1 followed by 100 zeros.
02

Define a Googolplex

A googolplex takes the concept of a googol even further. A googolplex is \(10^{\text{{googol}}}\), which means it is 1 followed by a googol (i.e., \(10^{100}\)) zeros. Therefore, a googolplex is \(10^{10^{100}}\).
03

Understand the Magnitude

The value of a googolplex is so vast that it is practically impossible to write out in its entirety. The universe does not contain enough matter to represent a googolplex in written form as it has more digits than there are atoms in the observable universe.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are essential tools in mathematics that help describe how many times a number is multiplied by itself. When using exponents, the expression can be significantly simplified. Let's break this down: if you see something like \(10^2\), it means 10 multiplied by itself twice, which equals 100. This becomes more useful when dealing with very large or small numbers. Instead of writing out long strings of zeros, exponents allow us to express these numbers succinctly. For example, \(10^{100}\) is much cleaner than writing out a 1 followed by 100 zeros.
  • Exponentiation makes it easier to understand and work with huge numbers.
  • It avoids errors that may arise from counting endless zeros.
  • It provides a standard format for expressing extremely large quantities.
Large Numbers
When discussing large numbers in mathematics, terms like a googol and googolplex represent mind-boggling quantities that are far beyond everyday usage. A googol was coined by a nine-year-old and represents \(10^{100}\), a 1 followed by a hundred zeros. Yet this is just the starting point!Now, imagine a googolplex, \(10^{(10^{100})}\), which takes this idea to a whole new unimaginable scale. It's 1 followed by a googol number of zeros. With large numbers:
  • They push the boundaries of what's conceivable by humanity.
  • They serve as examples for understanding the limitations of our numerical system.
  • They illustrate the beauty of mathematics, where imagination knows no bounds.
Number Magnitude
The astounding magnitude of numbers like a googolplex challenges our ability to fully grasp their size. Number magnitude helps us view how truly immense these figures are. Consider the universe's vastness, which contains approximately \(10^{80}\) atoms. Compared to a googolplex which has far more zeros, it's clear why writing a googolplex using atoms is impossible. It far exceeds not only our writing capabilities but also physical limits. Understanding number magnitude:
  • It assists us in comprehending astronomical and quantum concepts.
  • It highlights the limitations of human cognition regarding vast quantities.
  • It showcases the utility of mathematics in bridging abstract numbers with real-world scenarios.

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Most popular questions from this chapter

State the physical quantity corresponding to each of the following metric units. (a) \(\mathrm{m} / \mathrm{s}\) (b) \(\mathrm{g} / \mathrm{mL}\) (c) \(\operatorname{cal} /\left(g \times{ }^{\circ} \mathrm{C}\right)\) (d) \(\mathrm{g}\) solute \(/ 100-\mathrm{g}\) solution

Heating solid potassium chlorate with manganese dioxide catalyst produces solid potassium chloride and oxygen gas If \(455 \mathrm{~mL}\) of wet oxygen are collected over water at \(23{ }^{\circ} \mathrm{C}\) and \(766 \mathrm{~mm} \mathrm{Hg}\), what mass of potassium chlorate was decomposed? The vapor pressure of water at \(23^{\circ} \mathrm{C}\) is \(21 \mathrm{~mm} \mathrm{Hg}\)

Ground water sometimes contains traces of hydrogen sulfide, which has the odor of rotten eggs. Chlorine gas is used to purify the water for drinking. The resulting sulfur reacts with fluorine gas to give sulfur hexafluoride. The reactions are as follows: $$\begin{aligned}8 \mathrm{H}_{2} \mathrm{~S}(a q)+8 \mathrm{Cl}_{2}(g) & \longrightarrow 16 \mathrm{HCl}(a q)+\mathrm{S}_{8}(s) \\\\\mathrm{S}_{8}(s)+24 \mathrm{~F}_{2}(g) & \longrightarrow 8 \mathrm{SF}_{6}(g)\end{aligned} $$ Starting with \(0.950 \mathrm{~L}\) of \(\mathrm{Cl}_{2}\) (STP) and excess fluorine gas, calculate: (a) the mass of sulfur hexafluoride produced (b) the volume of sulfur hexafluoride produced at STP (c) the volume of aqueous \(0.0265 \mathrm{M} \mathrm{H}_{2} \mathrm{~S}\) that reacted

Given that 0.0142 mol of hydrogen chloride gas dissolves in \(47.5 \mathrm{~mL}\) of solution, draw a mole concept map and calculate each of the following: (a) grams of HCl gas (STP) dissolved in the solution (b) liters of HCl gas (STP) dissolved in the solution (c) molecules of HCl gas dissolved in the solution (d) molar concentration of the hydrochloric acid solution

State the basic metric system unit for each of the following physical quantities. (a) length (b) mass (c) volume (d) time (e) temperature (f) heat energy
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